
Associated Legendre hypergroups:
For \(\nu \in {R}_0\) , let
\(\gamma _n:=\frac{(\nu +1)_n}{2^n(\nu +\frac{1}{2})_n}\left(1+\sum _{k=1}^n \frac{\nu }{k+\nu }\right)\) ,
\(a_n:=\frac{\gamma _{n+1}}{\gamma _{n}}\) ,
\(b_n:=0\) , and \(c_n:=1-a_n\)  if \(n\ge 1\)  and \(\gamma _0=1\) .
The polynomial \(P_n\)  associated to the sequences
\((a_n)_{n\ge 1},(b_n)_{n\ge 1},(c_n)_{n\ge 1}\)  in the
recursion formula (REF )
is the \(n\) -th associated Legendre polynomial with
parameter \(\nu \) .
The Haar weights of the induced hypergroup on \({N}_0\)  are given
by \(h(0)=1\)  and
\(h(n)=\frac{2\nu +2n+1}{2\nu +1}\left( 1+ \sum _{k=1}^n\frac{\nu }{(k+\nu )^2}\right)^2 \) , \(n\ge 1\) ,
and the support of the Plancherel measure can be
identified with \([-1,1]\) ;
see [1]}.
If \(x\in (-1,1)\) ,
\(\pi (\lbrace \alpha _x\rbrace )=0\)  and \(\alpha _x\in C_0({N}_0)\) , so it follows from
Theorem REF 
that \({N}_0\)  is \(\alpha _x\) -amenable if and only if \(\alpha _x=1\) .

Pollaczek polynomials hypergroup: The Pollaczek polynomials
\(\lbrace P^{(\eta , \mu )}_n\rbrace _{n\in {N}_0}\)  depending
on the parameters \(\eta \ge 0\) , \(\mu >0\)  or \(-\frac{1}{2} <\eta <0\)  and
\(0\le \mu <\eta +\frac{1}{2}\)  induce a hypergroup structure on \({N}_0\)  [2]}.
The Haar weights are given by \(h(0)=1\)  and
\(h(n)=\frac{(2n+2\eta +2\mu +1)(2\eta +1)_n}{(2\eta +2\nu +1)n!}\left( \sum _{k=0}^n\left(\begin{array}{c}n \\k \\\end{array}\right)\frac{(2\mu )^k}{(2\eta +1)_k}\right)^2,\) 
and the Plancherel measure with the support
\(\mathcal {S}\cong [-1,1]\)  is given by \(d\pi (x)=A(x)dx\)  where
\(A(\cos t)=(\sin t)^{2\eta }|\Gamma (\eta +\frac{1}{2}+i\mu \cot (t))|^2\exp ((2t-\pi )\mu \cot (t))\) , \(0\le t\le \pi .\) 
Given \(x\in (-1,1)\) , since \(\pi (\lbrace \alpha _x\rbrace )=0\)  and \(\alpha _x\in C_0({N}_0)\) ,
by Theorem REF  we see that \({N}_0\)  is \(\alpha _x\) -amenable if and only if \(\alpha _x=1\) .

Generalized Soradi hypergroups: These are polynomial hypergroups
of type [V] on \({N}_0\)  [1]} with the characters
\(P_n(\cos \theta )= \frac{\sin (n+1)\theta - k\sin n\theta }{(nk+n+1)\sin \theta } \hspace{14.22636pt}(n\ge 1),\) 

and the density of the Plancherel measure on the dual space
\(\widehat{{N}}_0\cong [-1,1]\)  is given by
\(p(x):= \frac{2(1-x^2)^{1/2}}{\pi (1+k^2-2kx)} (k>1).\) 
For \(x\in [-1, 1)\) , since \(\pi (\lbrace \alpha _x\rbrace )=0\)  and
\(\alpha _x\in C_0({N}_0)\) ,
Theorem REF  implies that
\({N}_0\)  is \(\alpha _x\) -amenable if and only if \(\alpha _x=1\) .

Hypergroups associated with infinite distance-transitive graphs:
They are polynomial hypergroups on \({N}_0\)  depending on \(a, b\in {R}\)  with \(a, b\ge 2\) ; and, one can associate them with infinite distance-transitive graphs if \(a, b\)  are integers.
These hypergroups have been thoroughly studied by M.
Voit [4]}.
For \(b>a\ge 2\)  (see below) they provide a rare and interesting case of
\(\alpha \) -amenability of hypergroups. Their Haar weights and characters are given by
\(h^{(a,b)}(0):=1, \; h^{(a,b)}(n)=a (a-1)^{n-1} (b-1)^n\quad (n\ge 1),\) 

and
\(P_n^{(a,b)}(x) =\frac{a-1}{a\left( (a-1)(b-1)\right)^{n/2}} \left(U_n(x)+\frac{b-2}{\left( (a-1)(b-1)\right)^{1/2}}U_{n-1}(x)-\frac{1}{a-1}U_{n-2}(x) \right),\) 

respectively, where \(U_n(\cos t)=\frac{\sin (n+1)t}{\sin t}\)  are the
Tchebychev polynomials of the second kind and
\(u_{-1}=u_{-2}:=0\) . The dual space \(\widehat{{N}_0^{(a,b)}}\) 
can be identified with \([-s_1,s_1]\) , where \(s_1:= \frac{ab-a-b+1}{2\sqrt{(a-1)(b-1)}}\) .
The normalized orthogonality measure \(\pi \in M^1({R})\) 
is
\(d\pi (x)=A(x)dx|_{[-1,1]} \quad \quad \hspace{39.83368pt} \text{for}\;a\ge b\ge 2,\) 

and
\(d\pi (x)=A(x)dx|_{[-1,1]} +\frac{b-a}{b}ds_0 \quad \text{for}\; b>a\ge 2\) 

with \(A(x):= \frac{a}{2\pi }\frac{(1-x^2)^{1/2}}{(s_1-x)(x-s_0)}, \; s_0=\frac{2-a-b}{2\sqrt{(a-1)(b-1)}}\) .
Note that
\(P_n^{(a,b)} (s_1)=1\hspace{14.22636pt} \mbox{ and } \hspace{14.22636pt} P_n^{(a,b)}(s_0)=(1-b)^{-n} \hspace{14.22636pt} \mbox{ for } n\ge 0.\) 
Proposition 2.1 
Let \({N}_0^{(a,b)}\)  denote the above hypergroup. Then 
(i)
for \(a\ge b\ge 2\) , \({N}_0^{(a,b)}\) is \(\alpha _x\) -amenable
if and only if  \(x=s_1\) . 
(ii)
for \(b>a\ge 2\) , \({N}_0^{(a,b)}\) is \(\alpha _x\) -amenable if and only if
\(x=s_1\)  or \(x=s_0\) . 


(i) If \( x\in (-s_1, s_1)\) , then \(\pi (\lbrace \alpha _x\rbrace )=0\)  and
\(\alpha _x\in C_0({N}_0^{(a, b)})\) . So, applying Theorem
REF  yields that \({N}_0^{(a, b)}\)  is
\(\alpha _x\) - amenable if and only if \(x=s_0\) , as \(\alpha _{s_0}=1\) .
(ii) As in part (i), we can show that if \(x\ne s_0\) , then
\({N}_0^{(a, b)}\)  is \(\alpha _x\) - amenable if and only if \(x=s_1\) . In
the case of \(x=s_0\) , obviously \(\alpha _{s_0}\in \ell ^1({N}_0^{(a,b)})\) 
(see also [4]}) which implies, by Remark REF 
(ii), that
\({N}_0^{(a, b)}\)  is \(\alpha _{s_0}\) -amenable.
Remark 2.1.3 
Notice that in the previous example \(\widehat{K}\)  contains
two positive characters \(\alpha _{s_0}\)  and \(\alpha _{s_1}\)  with diverse behaviours. Indeed, Part (i) shows that \( {N}_0^{(a,b)}\) 
is \(\alpha _{s_1}\) -amenable but not \(\alpha _{s_0}\) -amenable if \(a\ge b \ge 2\) ,
whereas Part (ii) shows that
\( {N}_0^{(a,b)}\)  is \(\alpha _{s_1}\)  and \(\alpha _{s_0}\) -amenable for \(b>a\ge 2\) .
In latter case, the functional \(m_{\alpha _{s_0}}\) , given
in Remark REF  (ii), is a unique \(\alpha _{s_0}\) -mean on
\(\ell ^\infty ({N}_0^{(a, b)})\)  while
the cardinality of \(\alpha _{s_1}\) -means on \(\ell ^\infty ({N}_0^{(a, b)})\) 
is infinity; see [6]}, [7]}. 


